October  2021, 14(10): 3611-3628. doi: 10.3934/dcdss.2020431

Solving a class of biological HIV infection model of latently infected cells using heuristic approach

1. 

Department of Dermatology, Stomatology, Radiology and Physical Medicine, University of Murcia, Spain

2. 

Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

3. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain

4. 

Department of Electrical and Computer Engineering, COMSATS University, Islamabad, Attock Campus, Attock, Pakistan

* Corresponding author: Yolanda Guerrero–Sánchez

Received  September 2019 Revised  December 2019 Published  October 2021 Early access  November 2020

The intension of the recent study is to solve a class of biological nonlinear HIV infection model of latently infected CD4+T cells using feed-forward artificial neural networks, optimized with global search method, i.e. particle swarm optimization (PSO) and quick local search method, i.e. interior-point algorithms (IPA). An unsupervised error function is made based on the differential equations and initial conditions of the HIV infection model represented with latently infected CD4+T cells. For the correctness and reliability of the present scheme, comparison is made of the present results with the Adams numerical results. Moreover, statistical measures based on mean absolute deviation, Theil's inequality coefficient as well as root mean square error demonstrates the effectiveness, applicability and convergence of the designed scheme.

Citation: Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3611-3628. doi: 10.3934/dcdss.2020431
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M. Prague, Use of dynamical models for treatment optimization in HIV infected patients: A sequential Bayesian analysis approach, Journal de la Societe Francaise de Statistique, 157 (2016), 20.

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M. A. Z. RajaF. H. ShahM. Tariq and I. Ahmad, Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch's problem arising in plasma physics, Neural Computing and Applications, 29 (2018), 83-109.  doi: 10.1007/s00521-016-2530-2.

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[33]

M. A. Z. RajaU. FarooqN. I. ChaudharyA. M. Wazwaz and M. A.  , Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes, Applied Soft Computing, 38 (2016), 561-586.  doi: 10.1016/j.asoc.2015.10.015.

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show all references

References:
[1]

G. Adomian, Solving frontier problems modelled by nonlinear partial differential equations, Computers & Mathematics with Applications, 22 (1991), 91-94.  doi: 10.1016/0898-1221(91)90017-X.

[2]

I. Ahmad, et al., Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels, Neural Computing and Applications, 31 (2019), 9041-9059. doi: 10.1007/s00521-019-04203-y.

[3]

I. Ahmad, et al., Anticipated backward doubly stochastic differential equations with nonLiphschitz coefficients, Applied Mathematics and Nonlinear Sciences, 4 (2019), 9-20. doi: 10.1016/j.amc.2013.05.054.

[4]

S. Akbar, et al., Novel application of FO-DPSO for 2-D parameter estimation of electromagnetic plane waves, Neural Computing and Applications, 31 (2019), 3681-3690. doi: 10.1007/s00521-017-3318-8.

[5]

K. S. Al-Ghafri and H. Rezazadeh, Solitons and other solutions of (3+ 1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 289-304.  doi: 10.2478/AMNS.2019.2.00026.

[6]

N. Ali and G. Zaman, Asymptotic behavior of HIV-1 epidemic model with infinite distributed intracellular delays, Springer Plus, 5 (2016), 324. doi: 10.1186/s40064-016-1951-9.

[7]

N. Ali, G. Zaman and O. Algahtani, Stability analysis of HIV-1 model with multiple delays, Advances in Difference Equations, 2016 (2016), 88. doi: 10.1186/s13662-016-0808-4.

[8]

N. AliS. AhmadS. Aziz and G. Zaman, The Adomian decomposition method for solving HIV infection model of latently infected cells, Matrix Science Mathematic, 3 (2019), 5-8.  doi: 10.26480/msmk.01.2019.05.08.

[9]

J. Bleyer, Advances in the simulation of viscoplastic fluid flows using interior-point methods, Computer Methods in Applied Mechanics and Engineering, 330 (2018), 368-394.  doi: 10.1016/j.cma.2017.11.006.

[10]

D. W. Brzezinski, Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus, Applied Mathematics and Nonlinear Sciences, 3 (2018), 487-502.  doi: 10.2478/AMNS.2018.2.00038.

[11]

D. W. Brzezinski, Comparison of fractional order derivatives computational accuracy-right hand vs left hand definition, Applied Mathematics and Nonlinear Sciences, 2 (2017), 237-248.  doi: 10.21042/AMNS.2017.1.00020.

[12]

S. Effati and M. Pakdaman, Artificial neural network approach for solving fuzzy differential equations, Information Sciences, 180 (2010), 1434-1457.  doi: 10.1016/j.ins.2009.12.016.

[13]

A. P. Engelbrecht, Computational Intelligence: An Introduction, John Wiley & Sons, 2007. doi: 10.1002/9780470512517.ch1.

[14]

A. A. EsminR. A. Coelho and S. Matwin, A review on particle swarm optimization algorithm and its variants to clustering high-dimensional data, Artificial Intelligence Review, 44 (2015), 23-45.  doi: 10.1007/s10462-013-9400-4.

[15]

M. F. Fateh, et al., Differential evolution based computation intelligence solver for elliptic partial differential equations, Frontiers of Information Technology & Electronic Engineering, 20 (2019), 1445-1456. doi: 10.1631/FITEE.1900221.

[16]

M. GhoreishiA. M. Ismail and A. K. Alomari, Application of the homotopy analysis method for solving a model for HIV infection of CD4+ T-cells, Mathematical and Computer Modelling, 54 (2011), 3007-3015.  doi: 10.1016/j.mcm.2011.07.029.

[17]

K. Hattaf and N. Yousfi, Global properties of a discrete viral infection model with general incidence rate, Mathematical Methods in the Applied Sciences, 39 (2016), 998-1004.  doi: 10.1002/mma.3536.

[18]

K. Hattaf and N. Yousfi, A numerical method for a delayed viral infection model with general incidence rate, Journal of King Saud University-Science, 28 (2016), 368-374.  doi: 10.1007/s40435-015-0158-1.

[19]

K. Hattaf and N. Yousfi, Modeling the adaptive immunity and both modes of transmission in HIV infection, Computation, 6 (2018), 37. doi: 10.3390/computation6020037.

[20]

K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 21. doi: 10.3390/computation7020021.

[21]

W. HeY. Chen and Z. Yin, Adaptive neural network control of an uncertain robot with full-state constraints, IEEE transactions on cybernetics, 46 (2015), 620-629.  doi: 10.1109/TCYB.2015.2411285.

[22]

A. Khare and S. Rangnekar, A review of particle swarm optimization and its applications in solar photovoltaic system, Applied Soft Computing, 13 (2013), 2997-3006.  doi: 10.1016/j.asoc.2012.11.033.

[23]

D. MangoniA. TasoraA.   and R. Garziera, A primal-dual predictor-corrector interior point method for non-smooth contact dynamics, Computer Methods in Applied Mechanics and Engineering, 330 (2018), 351-367.  doi: 10.1016/j.cma.2017.10.030.

[24]

A. Mehmood, et al., Integrated intelligent computing paradigm for the dynamics of micropolar fluid flow with heat transfer in a permeable walled channel, Applied Soft Computing, 79 (2019), 139-162. doi: 10.1016/j.asoc.2019.03.026.

[25]

A. Mehmood, et al., Backtracking search heuristics for identification of electrical muscle stimulation models using Hammerstein structure, Applied Soft Computing, 84 (2019), 105705. doi: 10.1016/j.asoc.2019.105705.

[26]

S. Momani, Z. S. Abo-Hammour and O. M. Alsmadi, Solution of inverse kinematics problem using genetic algorithms, Applied Mathematics & Information Sciences, 10 (2016), 225. doi: 10.1016/j.ins.2014.03.128.

[27]

F. PelletierC. Masson and A. Tahan, Wind turbine power curve modelling using artificial neural network, Renewable Energy, 89 (2016), 207-214.  doi: 10.1016/j.renene.2015.11.065.

[28]

A. S. Perelson, Modeling the interaction of the immune system with HIV, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, Heidelberg, 1989,350–370. doi: 10.1007/978-3-642-93454-4_17.

[29]

A. S. PerelsonD. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells. Mathematical biosciences, Mathematical Biosciences, 114 (1993), 81-125. 

[30]

M. Prague, Use of dynamical models for treatment optimization in HIV infected patients: A sequential Bayesian analysis approach, Journal de la Societe Francaise de Statistique, 157 (2016), 20.

[31]

M. A. Z. RajaF. H. ShahM. Tariq and I. Ahmad, Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch's problem arising in plasma physics, Neural Computing and Applications, 29 (2018), 83-109.  doi: 10.1007/s00521-016-2530-2.

[32]

M. A. Z. RajaJ. A. Khan and T. Haroon, Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks, Journal of the Taiwan Institute of Chemical Engineers, 48 (2015), 26-39.  doi: 10.1016/j.jtice.2014.10.018.

[33]

M. A. Z. RajaU. FarooqN. I. ChaudharyA. M. Wazwaz and M. A.  , Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes, Applied Soft Computing, 38 (2016), 561-586.  doi: 10.1016/j.asoc.2015.10.015.

[34]

M. A. Z. RajaJ. MehmoodZ. SabirA. K. Nasab and M. A. Manzar, Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing, Neural Computing and Applications, 31 (2019), 793-812.  doi: 10.1007/s00521-017-3110-9.

[35]

M. A. Z. Raja, M. Umar, Z. Sabir, J. A. Khan and D. Baleanu, A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head, The European Physical Journal Plus, 133 (2018), 364. doi: 10.1140/epjp/i2018-12153-4.

[36]

M. A. Z. Raja, Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP, Connection Science, 26 (2014), 195-214.  doi: 10.1080/09540091.2014.907555.

[37]

M. A. Z. RajaM. S. AslamN. I. ChaudharyM. Nawaz and S. M. Shah, Design of hybrid nature-inspired heuristics with application to active noise control systems, Neural Computing and Applications, 31 (2019), 2563-2591.  doi: 10.1007/s00521-017-3214-2.

[38]

M. A. Z. RajaU. AhmedA. ZameerA. K. Kiani and N. I. Chaudhary, Bio-inspired heuristics hybrid with sequential quadratic programming and interior-point methods for reliable treatment of economic load dispatch problem, Neural Computing and Applications, 31 (2019), 447-475.  doi: 10.1007/s00521-017-3019-3.

[39]

M. A. Z. RajaM. S. AslamN. I. Chaudhary and W. U. Khan, Bio-inspired heuristics hybrid with interior-point method for active noise control systems without identification of secondary path, Frontiers of Information Technology & Electronic Engineering, 19 (2018), 246-259.  doi: 10.1631/FITEE.1601028.

[40]

E.S. Rosenberg, et al., Immune control of HIV-1 after early treatment of acute infection, Nature, 407 (2000), 523. doi: 10.1038/35035103.

[41]

Z. SabirM. A. ManzarM. A. Z. RajaM. Sheraz and A. M. Wazwaz, Neuro-heuristics for nonlinear singular Thomas-Fermi systems, Applied Soft Computing, 65 (2018), 152-169.  doi: 10.1016/j.asoc.2018.01.009.

[42]

Z. Sadegh and N. Miehran, A nonstandard finite difference scheme for solving fractional-order model of HIV-1 infection of CD4+t-cells, Iranian Journal of Mathematical Chemistry, 6 (2015), 169-184. 

[43]

J. C. Schaff, F. Gao, Y. Li, I. L. Novak and B. M. Slepchenko, Numerical approach to spatial deterministic-stochastic models arising in cell biology, PLoS Computational Biology, 12 (2016), 1005236. doi: 10.1371/journal.pcbi.1005236.

[44]

Y. Shi and R. C. Eberhart, Empirical study of particle swarm optimization, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, (Cat. No. 99TH8406), 3 (1999), 1945–1950). doi: 10.1109/CEC.1999.785511.

[45]

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Figure 1.  Graphical illustration of presented scheme for HIV infection model of latently infected cells
Figure 2.  Pseudo code using PSO-IPA
Figure 3.  Trained weights or decision variables of ANN on the basis of best fitness achieved
Figure 4.  Results for HIV infection spread model
Figure 5.  Comparative study on AE of the presented solutions using 5 neurons with the Adams results
Figure 6.  Analysis on MAD for convergence along with the histograms for 5 neurons
Figure 7.  Analysis on RMSE for convergence along with the histograms for 5 neurons
Figure 8.  Analysis on TIC for convergence along with the histograms for 5 neurons
Figure 9.  Statistics based results of Problem 1 for $ x(t) $ and $ w(t) $
Figure 10.  Statistics based results of Problem 1 for $ y(t) $ and $ \nu(t) $
Table 1.  List of parameter and setting used for reported study of HIV infection model
Index Description Settings [8]
$ S_{1} $ Initial value of uninfected CD4+T cells 7
$ S_{2} $ Initial value of infected CD4+T cells 2
$ S_{3} $ Initial value of Virus free cells 1
$ S_{4} $ Initial value of latently infected cells 4
$ \mu $ Rate of uninfected CD4+T cells 0.4
$ \lambda $ Recovery Rate of infected cells 0.3
$ d $ Death rate of uninfected CD4+T cells 0.01
$ \alpha $ Rate of infection spread 0.04
$ q $ Rate of removal of recombinants 0.1
$ a $ Death rate of virus free cells 0.2
$ u $ Death rate of latently infected cells 0.03
Index Description Settings [8]
$ S_{1} $ Initial value of uninfected CD4+T cells 7
$ S_{2} $ Initial value of infected CD4+T cells 2
$ S_{3} $ Initial value of Virus free cells 1
$ S_{4} $ Initial value of latently infected cells 4
$ \mu $ Rate of uninfected CD4+T cells 0.4
$ \lambda $ Recovery Rate of infected cells 0.3
$ d $ Death rate of uninfected CD4+T cells 0.01
$ \alpha $ Rate of infection spread 0.04
$ q $ Rate of removal of recombinants 0.1
$ a $ Death rate of virus free cells 0.2
$ u $ Death rate of latently infected cells 0.03
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